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  • On the tree-depth of random graphs

     Perarnau Llobet, Guillem; Serra Albo, Oriol
    Discrete applied mathematics
    Date of publication: 2014-05-11
    Journal article

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    Tree-depth is a parameter introduced under several names as a measure of sparsity of a graph. We compute asymptotic values of the tree-depth of a random graph on n vertices where each edge appears independently with probability p. For dense graphs, np -> +infinity, the tree-depth of a random graph G is aas td(G) = n - O ( root n/p). Random graphs with p = c/n, have aaslinear tree-depth when c > 1, the tree-depth is (-)(log n) when c = 1 and (-) (log log n) for c < 1. The result for c > 1 is derived from the computation of tree-width and provides a more direct proof of a conjecture by Gao on the linearity of tree-width recently proved by Lee, Lee and Oum (2012) [15]. We also show that, for c = 1, every width parameter is aasconstant, and that random regular graphs have linear tree-depth. (c) 2012 Elsevier B.V. All rights reserved.

  • Counting patterns in colored orthogonal arrays

     Montejano Cantoral, Amanda; Serra Albo, Oriol
    Discrete mathematics
    Date of publication: 2014-02-28
    Journal article

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    Let S be an orthogonal array OA(d, k) and let c be an r-coloring of its ground set X. We give a combinatorial identity which relates the number of vectors in S with given color patterns under c with the cardinalities of the color classes. Several applications of the identity are considered. Among them it is shown that every coloring of an orthogonal array OA(d, d - 1) contains a positive proportion of almost rainbow vectors and also of almost monochromatic vectors of every color. (C) 2013 Elsevier B.V. All rights reserved.

  • Yahya Ould Hamidoune's mathematical journey: a critical review of his work

     Plagne, Alain; Serra Albo, Oriol; Zemor, Gilles
    European journal of combinatorics
    Date of publication: 2013-11
    Journal article

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    We present the mathematical work of Yahya Ould Hamidoune, emphasizing his main achievements, notably in graph theory and additive combinatorics.

  • The Dicks-Ivanov problem and the Hamidoune problem

     Dicks McLay, Warren; Serra Albo, Oriol
    European journal of combinatorics
    Date of publication: 2013-11
    Journal article

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    We report on what we call the Hamidoune problem, inspired by a problem by Dicks and Ivanov. The problem asks if the inequality holds when A and B are finite subsets of a group G, each one with at least two elements, and A·2B denotes the set of elements which can be written in at least two different ways as a product of one element in A and one in B.

  • A structure theorem for small sumsets in nonabelian groups

     Serra Albo, Oriol; Zemor, Gilles
    European journal of combinatorics
    Date of publication: 2013-11
    Journal article

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    Let G be an arbitrary finite group and let S and T be two subsets such that |S| = 2, |T| = 2, and |T S| = |T| + |S| - 1 = |G| - 2. We show that if |S| = |G| - 4|G|1 / 2 then either S is a geometric progression or there exists a non-trivial subgroup H such that either |H S| = |S| + |H| - 1 or |S H| = |S| + |H| - 1. This extends to the nonabelian case classical results for abelian groups. When we remove the hypothesis |S| = |G| - 4|G|1 / 2 we show the existence of counterexamples to the above characterization whose structure is described precisely.

  • Inverse additive problems for Minkowski Sumsets II

     Freiman, Gregory A.; Grynkiewicz, David; Serra Albo, Oriol; Stanchescu, Y.V.
    Journal of geometric analysis
    Date of publication: 2013-01
    Journal article

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  • On the removal lemma for linear systems over Abelian groups

     Král, Daniel; Serra Albo, Oriol; Vena, Lluis
    European journal of combinatorics
    Date of publication: 2013-02
    Journal article

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  • Rainbow perfect matchings in complete bipartite graphs: existence and counting

     Perarnau Llobet, Guillem; Serra Albo, Oriol
    Combinatorics probability and computing
    Date of publication: 2013-09
    Journal article

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    A perfect matching M in an edge-coloured complete bipartite graph Kn,n is rainbow if no pair of edges in M have the same colour. We obtain asymptotic enumeration results for the number of rainbow perfect matchings in terms of the maximum number of occurrences of each colour. We also consider two natural models of random edge-colourings of Kn,n and show that if the number of colours is at least n, then there is with high probability a rainbow perfect matching. This in particular shows that almost every square matrix of order n in which every entry appears n times has a Latin transversal.

  • Random combinatorial structures with low dependencies: existence and enumeration.

     Perarnau Llobet, Guillem
    Defense's date: 2013-10-01
    Universitat Politècnica de Catalunya
    Theses

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    En aquesta tesi s'estudien diferents problemes en el camp de la combinatòria i la teoria de grafs, utilitzant el mètode probabilístic. Aquesta tècnica, introduïda per Erdös, ha esdevingut una eina molt potent per tal de donar proves existencials per certs problemes en diferents camps de les matemàtiques on altres mètodes no ho han aconseguit. Un dels seus principals objectius és l'estudi del comportament de les variables aleatòries. El cas en que aquestes variables compten el nombre d'esdeveniments dolents que tenen lloc en una estructura combinatòria és de particular interès. La idea del Paradigma de Poisson és estimar la probabilitat que tots aquests esdeveniments dolents no succeeixin a la vegada, quan les dependències entre ells són febles o escasses. En tal cas, aquesta probabilitat s'hauria de comportar de forma similar al cas on tots els esdeveniments són independents. El Lema Local de Lovász o la Desigualtat de Suen són exemples d'aquesta idea. L'objectiu de la tesi és estudiar aquestes tècniques ja sigui proveint-ne noves versions, refinant-ne les existents per casos particulars o donant-ne noves aplicacions. A continuació s'enumeren les principals contribucions de la tesi.La primera part d'aquesta tesi estén un resultat d'Erdös i Spencer sobre transversals llatins [1]. Els autors proven que qualsevol matriu d'enters on cap nombre apareix massa vegades, admet un transversal on tots els nombres són diferents. Això equival a estudiar els aparellaments multicolors en aresta-coloracions de grafs complets bipartits. Sota les mateixes hipòtesis que [1], es donen resultats sobre el nombre d'aquests aparellaments. En la segona part d'aquesta tesi s'estudien els codis identificadors. Un codi identificador és un conjunt de vèrtexs tal que tots els vèrtexs del graf tenen un veïnatge diferent en el codi. Aquí s'estableixen cotes en la mida d'un codi identificador mínim en funció dels graus i es resol parcialment una conjectura de Foucaud et al. [2]. En un altre capítol, es mostra que qualsevol graf suficientment dens conté un subgraf que admet un codi identificador òptim. En alguns casos, provar l'existència d'un cert objecte és trivial. Tot i així, es poden utilitzar les mateixes tècniques per obtenir resultats d'enumeració. L'estudi de patrons en permutacions n'és un bon exemple. A la tercera part de la tesi es desenvolupa una nova tècnica per tal d'estimar el nombre de permutacions d'una certa llargada que eviten còpies consecutives d'un patró donat. En particular, es donen cotes inferiors i superiors per a aquest nombre. Una de les conseqüències és la prova de la conjectura CMP enunciada per Elizalde i Noy [3] així com nous resultats en el comportament de la majoria dels patrons. En l'última part de la tesi s'estudia la Conjectura Lonely Runner. Aquesta coneguda conjectura diu que per qualsevol conjunt de corredors que corren al llarg d'un cercle unitari, hi ha un moment on tots els corredors estan suficientment lluny de origen centre. Aquí, es millora un resultat de Chen [4] ampliant la distància de tots els corredors a l'origen. També s'estén el teorema del corredor invisible de Czerwinski i Grytczuk [5].[1] P. Erdos and J. H. Spencer, Lopsided Lovász local lemma and Latin transversals, Discrete Appl. Math. 30 (1991), no. 2-3, 151¿154.[2] F. Foucaud, R. Klasing, A. Kosowski, and A. Raspaud, On the size of identifying codes in triangle-free graphs, Discrete Appl. Math. 160 (2012), no. 10-11, 1532¿1546.[3] S. Elizalde and M. Noy, Consecutive patterns in permutations, Adv. in Appl. Math. 30 (2003), 110¿125.[4] Y. G. Chen, View-obstruction problems in n-dimensional Euclidean space and a generalization of them, Acta Math. Sinica 37 (1994), no. 4, 551¿562.[5] S. Czerwinski and J. Grytczuk, Invisible runners in ¿nite ¿elds, Inform. Process. Lett. 108 (2008), no. 2, 64¿67.

  • On the cardinality of sumsets in torsion-free groups

     Böröczky, Károly J.; Pálfy, P.P.; Serra Albo, Oriol
    Bulletin of the London Mathematical Society
    Date of publication: 2012-10
    Journal article

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  • Rainbow-free 3-colorings of abelian groups

     Montejano Cantoral, Amanda; Serra Albo, Oriol
    Electronic journal of combinatorics
    Date of publication: 2012-02-23
    Journal article

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  • Remarks on the equality case of the Bonnesen inequality

     Böröczky, Károly J.; Serra Albo, Oriol
    Archiv der mathematik
    Date of publication: 2012-08
    Journal article

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  • A removal lemma for systems of linear equations over finite fields

     Král, Daniel; Serra Albo, Oriol; Vena, Lluis
    Israel journal of mathematics
    Date of publication: 2012-01
    Journal article

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  • Inverse additive problems for Minkowski sumsets I

     Freiman, Gregory A.; Grynkiewicz, David; Serra Albo, Oriol; Stanchescu, Y.V.
    Collectanea mathematica
    Date of publication: 2012-09
    Journal article

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  • Optimización y problemas extremales en teoria de grafos y combinatoria. Aplicacions a les redes de comunicación

     Aguilo Gost, Francisco de Asis Luis; Abiad Monge, Aida; Andrés Yebra, José Luis; Barriere Figueroa, Eulalia; Cámara Vallejo, Marc; Ball, Simeon Michael; Comellas Padro, Francesc de Paula; Dalfo Simo, Cristina; Fàbrega Canudas, Josep; Garriga Valle, Ernest; Gomez Marti, Jose; Llado Sanchez, Anna; López Masip, Susana Clara; Miralles De La Asuncion, Alicia; Mitjana Riera, Margarida; Muñoz Lopez, Francisco Javier; Pelayo Melero, Ignacio Manuel; Pérez Mansilla, Sonia; Rius Font, Miquel; Serra Albo, Oriol; Vena, Lluis; Vilaltella Castanyer, Joan; Zaragoza Monroig, Maria Luisa; Espona Dones, Margarida; Sau Valls, Ignasi; Montejano Cantoral, Amanda; Perarnau Llobet, Guillem; Moragas Vilarnau, Jordi; Vena Cros, Lluís; Andres Yebra, Jose Luis; Fiol Mora, Miquel Àngel
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  • Yahya Ould Hamidoune, 1948--2011

     Llado Sanchez, Anna; Serra Albo, Oriol; Plagne, Alain
    Date: 2012-03-01
    Report

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  • The Removal Lemma: algebraic versions and applications

     Vena Cros, Lluís
    Defense's date: 2012-07-02
    Department of Applied Mathematics IV, Universitat Politècnica de Catalunya
    Theses

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  • Rainbow Matchings: Existence and counting

     Perarnau Llobet, Guillem; Serra Albo, Oriol
    Electronic notes in discrete mathematics
    Date of publication: 2011-12-01
    Journal article

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  • On the number of monochromatic solutions of integer linear systems on Abelian groups

     Serra Albo, Oriol; Vena, Lluis
    Electronic notes in discrete mathematics
    Date of publication: 2011-12-01
    Journal article

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    Let G be a finite cyclic group an let r be a positive integer. Let A be a k×m matrix with integer entries. We show that, if A satisfies some natural conditions, then the homogeneous linear system Ax=0 has Ω(|GN|m−k) monochromatic solutions for each r-coloring of GN\{0} and sufficiently large N. Density versions of this counting result are also addressed.

  • Properties of two-dimensional sets with small sumset

     Grynkiewicz, D; Serra Albo, Oriol
    Journal of combinatorial theory. Series A
    Date of publication: 2010-02
    Journal article

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  • INTERNATIONAL WORKSHOP ON OPTIMAL NETWORK TOPOLOGIES

     Balbuena Martinez, Maria Camino Teofila; Mitjana Riera, Margarida; Serra Albo, Oriol; Fàbrega Canudas, Josep; Comellas Padro, Francesc de Paula
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  • On the tree depth of random graphs

     Perarnau Llobet, Guillem; Serra Albo, Oriol
    Jornadas de Matemática Discreta y Algorítmica
    Presentation's date: 2010-06-15
    Presentation of work at congresses

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    The tree-depth of a graph G is a parameter that plays a crucial role in the theory of bounded expansion classes and has been introduced under numerous names. We describe the asymptotic behaviour of this parameter in the case of dense and of sparse random graphs. The results also provide analogous descriptions for the tree-width of random graphs.

  • Asymptotic upper bounds on the generalized chromatic numbers in classes of graphs of bounded degree

     Perarnau Llobet, Guillem; Serra Albo, Oriol
    French Combinatorial Conference
    Presentation's date: 2010-07
    Presentation of work at congresses

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  • On the number of solutions of regular systems in sets with positive density in abelian groups

     Král, Daniel; Serra Albo, Oriol; Vena, Lluis
    Jornadas de Matemática Discreta y Algorítmica
    Presentation's date: 2010-06-16
    Presentation of work at congresses

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  • A mathematical model for dynamic memory networks  Open access

     Fàbrega Canudas, Josep; Fiol Mora, Miquel Àngel; Serra Albo, Oriol; Andrés Yebra, José Luis
    International Workshop on Optimal Network Topologies
    Presentation's date: 2010-06-11
    Presentation of work at congresses

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    The aim of this paper is to bring together the work done several years ago by M.A. Fiol and the other authors to formulate a quite general mathematical model for a kind of permutation networks known as dynamic memories. A dynamic memory is constituted by an array of cells, each storing one datum, and an interconnection network between the cells that allows the constant circulation of the stored data. The objective is to design the interconnection network in order to have short access time and a simple memory control. We review how most of the proposals of dynamic memories that have appeared in the literature fit in this general model, and how it can be used to design new structures with good access properties. Moreover, using the idea of projecting a digraph onto a de Bruijn digraph, we propose new structures for dynamic memories with vectorial capabilities. Some of these new proposals are based on iterated line digraphs, which have been widely and successfully used by M.A. Fiol and his coauthors to solve many different problems in graph theory.

  • On the chromatic number of circulant graphs

     Barajas Tomas, Javier; Serra Albo, Oriol
    Discrete mathematics
    Date of publication: 2009-09
    Journal article

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    Rainbow-free 3-colorings in abelian groups  Open access

     Montejano Cantoral, Amanda; Serra Albo, Oriol
    Electronic notes in discrete mathematics
    Date of publication: 2009-07-27
    Journal article

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    A 3–coloring of an abelian group G is rainbow–free if there is no 3–term arithmetic progression with its members having pairwise distinct colors. We describe the structure of rainbow–free colorings of abelian groups. This structural description proves a conjecture of Jungi´c et al. on the size of the smallest chromatic class of a rainbow–free coloring of cyclic groups.

  • Cycle codes of graphs and MDS array codes

     Serra Albo, Oriol; Zemor, Gilles
    Electronic notes in discrete mathematics
    Date of publication: 2009-07-30
    Journal article

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    We investigate how to colour edges of graph so that any two colours make up a spanning tree. This problem is equivalent to transforming the cycle code of a graph into a Maximum Distance Separable (MDS) array code. Adopting this graph-theoretical interpretation allows us to provide a compact description of some families of low density MDS array codes in terms of cycle codes of coloured graphs. This includes a short description of Xu et al.’s “B-code”, together with its erasure and error decoding algorithms. We also give a partial answer to Xu et al.’s question about efficient error decoding for the dual B-code. We give alternative families of MDS array cycle codes, and in passing prove that an optimal MDS array cycle code of minimum column distance 4 is given by an appropriate colouring of the Petersen graph. We introduce double array colourings which allow the decoding of column or row errors and provides a new interesting graph theoretical problem. We give infinite families of graphs which admit double array colourings.

  • Connectivity of addition Cayley graphs

     Serra Albo, Oriol
    Journal of combinatorial theory. Series A
    Date of publication: 2009-01
    Journal article

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  • Special Issue on EUROCOMB'07: COMBINATORICS, GRAPH THEORY AND APPLICATIONS Preface

     Marquez, A; Nesetril, J; Serra Albo, Oriol
    European journal of combinatorics
    Date of publication: 2009-10
    Journal article

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  • A combinatorial proof of the Removal Lemma for Groups

     Kral, D; Serra Albo, Oriol; Vena, L
    Journal of combinatorial theory. Series A
    Date of publication: 2009-05
    Journal article

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    Punctured combinatorial Nullstellensätze  Open access

     Ball, Simeon Michael; Serra Albo, Oriol
    Combinatorica
    Date of publication: 2009
    Journal article

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    In this article we present a punctured version of Alon's Nullstellensatz which states that if $f$ vanishes at nearly all, but not all, of the common zeros of some polynomials $g_1(X_1),\ldots,g_n(X_n)$ then every $I$-residue of $f$, where the ideal $I=\langle g_1,\ldots,g_n\rangle$, has a large degree. Furthermore, we extend Alon's Nullstellensatz to functions which have multiple zeros at the common zeros of $g_1,g_2,\ldots,g_n$ and prove a punctured version of this generalised version. Some applications of these punctured Nullstellens\"atze to projective and affine geometries over an arbitrary field are considered which, in the case that the field is finite, will lead to some bounds related to linear codes containing the all one vector.

  • Large sets with small doubling modulo $p$ are well covered by an arithmetic progression

     Serra Albo, Oriol; Zémor, Gilles
    Annales de l'Institut Fourier
    Date of publication: 2009-10-05
    Journal article

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    On sums of dilates  Open access

     Javier, Cilleruelo; Hamidoune, Yahya O.; Serra Albo, Oriol
    Combinatorics probability and computing
    Date of publication: 2009-10-05
    Journal article

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    For k prime and A a finite set of integers with |A| ≥ $3(k-1)^2$ (k-1)! we prove that |A+k·A| ≥ (k+1)|A| − ⌈k(k+2)/4⌉ where k·A = {ka:a∈A}. We also describe the sets for which equality holds.

  • Colored combinatorial structures: homomorphisms and counting.

     Montejano Cantoral, Amanda
    Defense's date: 2009-06-09
    School of Mathematics and Statistics (FME), Universitat Politècnica de Catalunya
    Theses

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  • Investigando las matemáticas de las redes de comunicación

     Fàbrega Canudas, Josep; Serra Albo, Oriol
    Date: 2009-07-13
    Report

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  • SOME GEOMETRIC ASPECTS OF SUM SETS

     Serra Albo, Oriol
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  • COMBINATÒRIA , TEORIA DE GRAFS I APLICACIONS

     Comellas Padro, Francesc de Paula; López Masip, Susana Clara; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest; Espona Dones, Margarida; Rius Font, Miquel; Aroca Farrerons, Josep Maria; Aguilo Gost, Francisco de Asis Luis; Dalfo Simo, Cristina; Moragas Vilarnau, Jordi; Cámara Vallejo, Marc; Gago Alvarez, Silvia; Ball, Simeon Michael; Andres Yebra, Jose Luis; Pérez Mansilla, Sonia; Mitjana Riera, Margarida; Llado Sanchez, Anna; Barriere Figueroa, Eulalia; Fàbrega Canudas, Josep; Muñoz Lopez, Francisco Javier; Zaragoza Monroig, Maria Luisa; Gomez Marti, Jose; Miralles De La Asuncion, Alicia; Pelayo Melero, Ignacio Manuel; Serra Albo, Oriol
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  • PROBLEMAS EXTREMALES Y DE OPTIMIZACIÓN EN TEORIA DE GRAFOS Y COMBINATORIA: APLICACIÓN AL ANALISIS Y ALGORITMOS DE REDES DE COMUNICAC

     Abiad Monge, Aida; Andres Yebra, Jose Luis; Aguilo Gost, Francisco de Asis Luis; Aroca Farrerons, Josep Maria; Ball, Simeon Michael; Barajas Tomas, Javier; Barguilla Navarrete, Jorge; Barriere Figueroa, Eulalia; Cámara Vallejo, Marc; Comellas Padro, Francesc de Paula; Dalfo Simo, Cristina; Espona Dones, Margarida; Fàbrega Canudas, Josep; Gago Alvarez, Silvia; Garriga Valle, Ernest; Gomez Marti, Jose; Llado Sanchez, Anna; López Masip, Susana Clara; Miralles De La Asuncion, Alicia; Mitjana Riera, Margarida; Montejano Cantoral, Amanda; Moragas Vilarnau, Jordi; Muñoz Lopez, Francisco Javier; Pelayo Melero, Ignacio Manuel; Perarnau Llobet, Guillem; Pérez Mansilla, Sonia; Rius Font, Miquel; Sau Valls, Ignasi; Serra Albo, Oriol; Vena, Lluis; Vilaltella Castanyer, Joan; Zaragoza Monroig, Maria Luisa; Vena Cros, Lluís; Fiol Mora, Miquel Àngel
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  • On some subgroup chains related to Kneser's theorem

     Serra Albo, Oriol
    Journal de théorie des nombres de Bordeaux
    Date of publication: 2008-10
    Journal article

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  • The lonely runner with seven runners

     Serra Albo, Oriol
    Electronic journal of combinatorics
    Date of publication: 2008-05
    Journal article

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  • On s-arc transitive hypergraphs

     Pérez Mansilla, Sonia; Serra Albo, Oriol
    European journal of combinatorics
    Date of publication: 2008-05
    Journal article

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  • Homomorphisms: Structure and highlights - Preface

     Nesetril, J; Serra Albo, Oriol
    European journal of combinatorics
    Date of publication: 2008-05
    Journal article

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  • On the critical pair theory in abelian groups: beyond Chowla's theorem

     Serra Albo, Oriol
    Combinatorica
    Date of publication: 2008-07
    Journal article

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  • On complete subsets of the cyclic group

     Hamidoune, Yahya O.; Llado Sanchez, Anna; Serra Albo, Oriol
    Journal of combinatorial theory. Series A
    Date of publication: 2008-10
    Journal article

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  • Array cycle codes

     Serra Albo, Oriol
    Second Workshop on Mathematical Cryptology
    Presentation's date: 2008-10-25
    Presentation of work at congresses

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  • Chairman Conferència Convidada

     Serra Albo, Oriol
    Building Bridges
    Presentation's date: 2008-08-05
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  • Organitzador Minisymposia Combstru

     Serra Albo, Oriol
    European Congress of Mathematics
    Presentation's date: 2008-07-14
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  • A removal lemma for linear systems

     Serra Albo, Oriol
    Jornadas de Matemática Discreta y Algorítmica
    Presentation of work at congresses

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  • Punctured Combinatorial Nullstellensatz

     Ball, Simeon Michael; Serra Albo, Oriol
    Jornadas de Matemática Discreta y Algorítmica
    Presentation's date: 2008-07-21
    Presentation of work at congresses

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